\newproblem{lay:4_1_11}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.1.11}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Ana Pe\~na Gil, Jan. 19th 2014} \\}{}

  % Problem statement
	Let $W$ be the set of all vectors of the form $\begin{pmatrix}2b+3c \\ -b \\ 2c\end{pmatrix}$, 
	where $b$ and $c$ are arbitrary. Find the vectors $\mathbf{u}$ and $\mathbf{v}$ such that
	$H = \mathrm{Span}\{\mathbf{u}, \mathbf{v})\}$. Why does this show that $W$ is a subspace of
	$\mathbb{R}^3$?
}{
   % Solution
	Let $\mathbf{w}\in W$. We can write \[\mathbf{w}=\begin{pmatrix}2b+3c \\ -b \\ 2c\end{pmatrix}
	=b\begin{pmatrix}2 \\ -1 \\ 0\end{pmatrix} + c \begin{pmatrix}3 \\ 0 \\ 2\end{pmatrix}
	 = b\mathbf{u} + c\mathbf{v}\]
	So, $H = \mathrm{Span}\{(\mathbf{u}, \mathbf{v})\}=\mathrm{Span}\{(2, -1, 0), (3, 0, 2)\}$. 
	$W$ is a vector subspace $\mathbb{R}^3$ because it can be generated by a set of vector of
	$\mathbb{R}^3$.
	
}
\useproblem{lay:4_1_11}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
